1. Key Concept: General Term in Binomial Expansion

The binomial theorem states that for any real numbers $a$ and $b$, and any non-negative integer $n$, the expansion of $(a+b)^n$ is given by: $(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$ The $(r+1)^{th}$ term, often denoted as $T_{r+1}$, in this expansion is given by the formula: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ When the binomial is of the form $(a-b)^n$, it can be written as $(a + (-b))^n$. In this case, the general term becomes: $T_{r+1} = \binom{n}{r} a^{n-r} (-b)^r = (-1)^r \binom{n}{r} a^{n-r} b^r$ This formula is crucial for finding any specific term in the expansion without writing out the entire series.

2. Identifying Terms and Applying the Formula

We are given the binomial expansion of ${\left( {{x \over 4} - {{12} \over {{x^2}}}} \right)^{12}}$ Comparing this to $(a-b)^n$:

• $a = \frac{x}{4}$
• $b = \frac{12}{x^2}$
• $n = 12$

Now, we substitute these into the general term formula $T_{r+1} = (-1)^r \binom{n}{r} a^{n-r} b^r$: $T_{r+1} = (-1)^r \binom{12}{r} \left(\frac{x}{4}\right)^{12-r} \left(\frac{12}{x^2}\right)^r$ Explanation: We apply the general term formula directly, substituting the specific parts of our given expression. The $(-1)^r$ accounts for the alternating signs due to the minus sign in the binomial.

3. Isolating the Power of x

To find the term independent of $x$, we need to separate the terms involving $x$ from the constant terms. $T_{r+1} = (-1)^r \binom{12}{r} \frac{x^{12-r}}{4^{12-r}} \frac{12^r}{(x^2)^r}$ $T_{r+1} = (-1)^r \binom{12}{r} \frac{x^{12-r}}{4^{12-r}} \frac{12^r}{x^{2r}}$ Now, combine the powers of $x$: $x^{12-r} \cdot x^{-2r} = x^{12-r-2r} = x^{12-3r}$. And combine the constant terms: $\frac{1}{4^{12-r}} \cdot 12^r$. $T_{r+1} = (-1)^r \binom{12}{r} \left(\frac{1}{4}\right)^{12-r} (12)^r x^{12-3r}$ Explanation: We distribute the exponents to individual factors and use exponent rules ($(x^m)^n = x^{mn}$ and $x^m / x^n = x^{m-n}$) to simplify the expression, especially to consolidate all powers of $x$ into a single term. This step is crucial for identifying the exponent of $x$.

4. Finding the Term Independent of x

A term is independent of $x$ if the power of $x$ in that term is zero. From the previous step, the power of $x$ is $12-3r$. So, we set this exponent equal to zero: $12 - 3r = 0$ $3r = 12$ $r = 4$ Explanation: The definition of a term "independent of x" means that the variable x does not appear in that term, which mathematically translates to x being raised to the power of 0 ($x^0 = 1$). Solving for $r$ tells us which specific term in the expansion satisfies this condition.

5. Calculating the Specific Term

Now that we have $r=4$, we can substitute this value back into the expression for $T_{r+1}$ to find the $T_5$ term (since $r+1 = 4+1 = 5$). $T_5 = (-1)^4 \binom{12}{4} \left(\frac{1}{4}\right)^{12-4} (12)^4$ $T_5 = (1) \binom{12}{4} \left(\frac{1}{4}\right)^8 (12)^4$ First, calculate the binomial coefficient $\binom{12}{4}$: $\binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4!8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1}$ $\binom{12}{4} = \frac{11880}{24} = 495$ Next, simplify the powers of 4 and 12: $(12)^4 = (3 \times 4)^4 = 3^4 \times 4^4$ Substitute these values back into the expression for $T_5$: $T_5 = 495 \cdot \frac{1}{4^8} \cdot (3^4 \cdot 4^4)$ $T_5 = 495 \cdot \frac{3^4 \cdot 4^4}{4^8}$ $T_5 = 495 \cdot \frac{3^4}{4^{8-4}}$ $T_5 = 495 \cdot \frac{3^4}{4^4}$ Explanation: We substitute the found value of $r$ to pinpoint the exact term. We then calculate the numerical coefficient $\binom{12}{4}$ and simplify the powers of the constants. Expressing $12^4$ as $(3 \times 4)^4$ is a key step to align with the target format, which involves powers of 3 and 4. This simplification reduces the powers of 4.

6. Equating and Solving for k

The problem states that the term independent of $x$ is $\left( {{{{3^6}} \over {{4^4}}}} \right)k$. We found the term independent of $x$ to be $T_5 = 495 \cdot \frac{3^4}{4^4}$. So, we can set up the equation: $495 \cdot \frac{3^4}{4^4} = \frac{3^6}{4^4} \cdot k$ To solve for $k$, we first cancel out the common $4^4$ term from both sides: $495 \cdot 3^4 = 3^6 \cdot k$ Now, isolate $k$: $k = \frac{495 \cdot 3^4}{3^6}$ Using exponent rules ($x^m / x^n = x^{m-n}$): $k = \frac{495}{3^{6-4}}$ $k = \frac{495}{3^2}$ $k = \frac{495}{9}$ $k = 55$ Explanation: We equate our calculated term to the given form involving $k$. This allows us to set up an algebraic equation. Simplification of powers of 3 and performing the division leads directly to the value of $k$. Notice that $495 = 9 \times 55$, which further confirms the calculation.

7. Tips for Success and Common Mistakes

• Sign Convention: Be careful with the sign in the binomial expansion. If it's $(a-b)^n$, remember the $(-1)^r$ factor in the general term formula.
• Exponent Rules: Master exponent rules, especially when dealing with variables in the denominator (e.g., $1/x^2 = x^{-2}$) and distributing powers $(AB)^n = A^n B^n$.
• Simplification: Fully simplify numerical coefficients and powers, especially by breaking down composite numbers into their prime factors (e.g., $12 = 3 \times 4$). This makes it easier to match with the desired format of the answer.
• Term vs. Coefficient: Understand the difference between the "term" (which includes the $x$ part) and the "coefficient" (the numerical part). Here, we needed the entire term independent of $x$ to equate it correctly.

8. Summary/Key Takeaway

To find a term independent of $x$ (or any variable) in a binomial expansion:

1. Write down the general term $T_{r+1}$.
2. Simplify the general term to clearly identify the exponent of the variable.
3. Set the exponent of the variable to zero and solve for $r$.
4. Substitute the value of $r$ back into the general term (excluding the variable part) to find the numerical coefficient of that specific term.
5. If the problem provides a specific form for the term, carefully simplify your result to match that form and solve for any unknown constants like $k$.

The final answer is $\boxed{55}$.